CAT - Question from Pipes and Cisterns

Question

There are n pipes that fill a tank. Pipe 1 can fill the tank in 2 hours, Pipe 2 in 3 hours, Pipe 3 in 4 hours and so on. Pipe 1 is kept open for 1 hour, pipe 2 for 1 hour, then pipe 3 and so on. In how many hours will the tank get filled completely?

Explanation

Almost all questions can be approached well by asking the question "What happens in 1 hour" (or 1 day, or 1 minutes)

So, let us start with that

In 1 hour, pipe 1 fills 1/2 of the tank. So, in the first hour, the tank will not be filled
In 1 hour, pipe 2 fills 1/3 of the tank. So, in two hours we would have filled 1/2 + 1/3 of the tank, or 5/6 of the tank. So, by the end of the second hour, the tank would still not be filled.
Let us move to the third hour. In 1 hour, pipe 3 fills 1/4 of the tank. So, by the end of the third hour, we should have filled 5/6 + 1/4 = 13/12 of the tank.

Oops, one cannot fill 13/12 of a tank. What this tells us is that the tank gets filled in the 3rd hour. 

When exactly during the third hour?

At the beginning of the third hour, we still have 1/6 of the tank still to fill. Pipe 3 can fill at the rate of 1/4 of the tank per hour. Or, pipe 3 will take 2/3 hours to fill the tank.

Or, the tank will be filled in 2 hours and 40 minutes.

This is an example of a sequence that is a harmonic progression. The formulae for HP are needlessly confusing. So, simple step-by-step approach works best. 

Pipes Cisterns - Solutions

Have given below solutions to the questions on Pipes Cisterns

1) Pipe A, B and C are kept open and together fill a tank in t minutes. Pipe A is kept open throughout, pipe B is kept open for the first 10 minutes and then closed. Two minutes after pipe B is closed, pipe C is opened and is kept open till the tank is full. Each pipe fills an equal share of the tank. Furthermore, it is known that if pipe A and B are kept open continuously, the tank would be filled completely in t minutes. Find t?

A is kept open for all t minutes and fills one-third the tank. Or, A should be able to fill the entire tank in '3t' minutes.

A and B together can fill the tank completely in t minutes. A alone can fill it in 3t minutes.
A and B together can fill 1/t of the tank in a minute. A alone can fill 1/3t of the tank in a minute. So, in a minute, B can fill 1/t - 1/3t = 2/3t. Or, B takes 3t/2 minutes to fill an entire tank

To fill one-third the tank, B will take t/2 minutes. B is kept open for t - 10 minutes.

t/2 = t - 10, t = 20 minutes.

A takes 60 mins to fill the entire tank, B takes 30 minutes to fill the entire tank. A is kept open for all 20 minutes. B is kept open for 10 minutes.

C, which is kept open for 8 minutes also fills one-third the tank. Or, c alone can fill the tank in 24 minutes.


2) Pipe A fills a tank at the rate of 100lit/min, Pipe B fills at the rate of 25 lit/min, pipe C drains at the rate of 50 lit/min. The three pipes are kept open for one minute each, one after the other. If the capacity of the tank is 7000 liters, how long will it take to fill the tank if

i) A is kept open first, followed by B and then C
Each cycle of 3 minutes, 75 liters get filled. 100 + 25 - 50. So, after 3 minutes the tank would have 75 liters
6 mins - 150 liters
9 mins - 225 liters
30 mins - 750 liters
270 mins - 6750 liters
273 mins - 6825 liters
279 mins - 6975 liters

In the 280th minute, pipe A would be open and it would fill the remaining 25 liters in 15 seconds. So, it would take 279 mins and 15 seconds to fill the tank.

The most important thing in these type of questions is to think in terms of cycles till we reach close to the required target and then think in simple steps.

ii) B first, followed by A, and then C

Each cycle of 3 minutes, 75 liters get filled. 25 + 100 - 50. So, after 3 minutes the tank would have 75 liters
6 mins - 150 liters
9 mins - 225 liters
30 mins - 750 liters
270 mins - 6750 liters
273 mins - 6825 liters
279 mins - 6975 liters

In the 280th minute, pipe B would be open and it would fill the remaining 25 liters in one minutes. So, it would take 280 mins.

iii) B first, followed by C, and then A
In this case also, Each cycle of 3 minutes, 75 liters get filled. 25 -50 + 100. But there is a small catch here. In the first set of 3 minutes, we would fill up to about 100 liters. After 1 minute, we would be at 25 liters, after 2 minutes, we would be at 0 liters and in the third minute, the tank would be 100 liters full.

6 mins - 170 liters
9 mins - 250 liters
30 mins - 775 liters
270 mins - 6775 liters
273 mins - 6850 liters
276 mins - 6925 liters
279 minutes - 7000 liters

So, it would take 279 mins and 15 seconds to fill the tank.

The most important thing in these type of questions is to think in terms of cycles till we reach close to the required target and then think in simple steps.

3) 4 men and 6 women complete a task in 24 days. If the women are at least half as efficient as the men, but not more efficient than the men, what is the range of the number of days for 6 women and 2 men to complete a task twice as difficult as this one?

4m and 6w finish in 24 days.
In one day, 4m + 6w = 1/24 of task

In these questions, just substitute extreme values to get the whole range
If a woman is half as efficient as man
4m + 3m = 1/24, 7m = 1/24, m = 1/168
6w + 2m = 3m + 2m = 5m, 5m will take 168/5 days = 33.6 days

If a woman is as efficient as a man
4m + 6w finish in 24 days
10m finish 1/24 of task in a day
6w + 2m = 8m, 8m will take 240/8 = 30 days to finish the task.

So, the range = 30 to 33.6 days. The new team will take 30 to 33.6 days to finish the task.

4) A fill pipe can fill a tank in 20 hours, a drain pipe can drain a tank in 30 hours. If a system of n pipes (fill pipes and drain pipes put together) can fill the tank in exactly 5 hours, which of the following are possible values of n (More than one option could be correct)?
1) 32 2) 54 3) 29 4) 40

3 fill pipes cancel out 2 drain pipes. Plus, you need an additional 4 fill pipes fill the tank in 5 hours. so the answer has to be 5k + 4. Both 54 and 29 are possible.

Pipes Cisterns - Questions

Have given below a few questions on pipes and cisterns.

1) Pipe A, B and C are kept open and together fill a tank in t minutes. Pipe A is kept open throughout, pipe B is kept open for the first 10 minutes and then closed. Two minutes after pipe B is closed, pipe C is opened and is kept open till the tank is full. Each pipe fills an equal share of the tank. Furthermore, it is known that if pipe A and B are kept open continuously, the tank would be filled completely in t minutes. Find t?

2) Pipe A fills a tank at the rate of 100lit/min, Pipe B fills at the rate of 25 lit/min, pipe C drains at the rate of 50 lit/min. The three pipes are kept open for one minute each, one after the other. If the capacity of the tank is 7000 liters, how long will it take to fill the tank if

i) A is kept open first, followed by B and then C
ii) B first, followed by A, and then C
iii) B first, followed by C, and then A

3) 4 men and 6 women complete a task in 24 days. If the women are at least half as efficient as the men, but not more efficient than the men, what is the range of the number of days for 6 women and 2 men to complete a task twice as difficult as this one?

4) A fill pipe can fill a tank in 20 hours, a drain pipe can drain a tank in 30 hours. If a system of n pipes (fill pipes and drain pipes put together) can fill the tank in exactly 5 hours, which of the following are possible values of n (More than one option could be correct)?
1) 32 2) 54 3) 29 4) 40

CAT Pipes and Cisterns - Basic Questions

Have given below a few basic questions on Pipes and Cisterns.

1. Pipe A can fill a tank in 6 hours. On account of a crack at the bottom of the tank through which water leaks, it takes an hour more for the tank to overflow. How long will the crack take to empty a full tank when pipe A is kept shut?

2. Pipe A that can fill a tank in 40 minutes and pipe B that can fill a tank in 20 minutes are opened when the tank is empty. Pipe B is shut 16 minutes before the tank overflows. How long did it take for the tank to overflow?

3. Pipe A can fill a tank in 36 minutes. Pipe B can fill the same tank in 72 minutes. There is a drain pipe, pipe C, at the bottom of the tank. A man opens pipe A and pipe B when the tank is empty and returns back to shut them when the tank should overflow. However, he finds that he had kept pipe C also open and shuts it immediately. The tank overflows in the next 6 minutes. How long will pipe C take to drain the tank, if pipes A and B are kept shut?

4. A tank is fitted with 22 pipes. Some of these are fill pipes that fill the tank and others waste pipes that drain the tank. If each fill pipe can fill the tank in 24 hours and each drain pipe can drain the tank in 16 hours, and if all the pipes are kept open simultaneously when the tank is empty, the tank overflows in 12 hours. How many of these are waste pipes?

5. Adrain completes 60% of a task in 72 days. He realizes that to meet his deadline, he needs additional help and gets Allen who is thrice as efficient as Adrain to help him. They complete the task with four days to spare. What was the deadline for the project from the day Adrain started work on it?

6. Working together A and B can complete a task in 18 days. If A worked alone, completed the first half of the task and then B took over from A, the task will be completed in 48 days. If A is more efficient than B, how long will A alone take to complete the task?

7. Working alone, Shiv will take 27 more hours to complete a task than if he worked with George to complete the task. Working alone, George will take 3 more hours to complete the task than if he worked with Shiv. How long will George take to complete the task, if he worked alone?

8. 5 men and 7 women can complete a task in 12 days. However, if 11 men and 9 women worked on the same task, the work will be completed only in 8 days. How long will a man and woman take to complete the task?